%%%%%%%%%%%%%%%%%%%%%%%%
%%
%% THIS IS A LATEX FILE
%%
%% PLEASE RUN IT TWICE BEFORE YOU PRINT IT
%%
%% The output consists of 10 pages
%%
%%%%%%%%%%%%%%%%%%%%%%%%%
\documentstyle[12pt]{article}
\parskip=1ex
\parindent=1.5em
\textwidth=15.25cm
\textheight=23cm
\evensidemargin=0.71cm
\oddsidemargin=0.2cm
\topmargin=-1.5cm
\baselineskip=18pt
%\renewcommand{\baselinestretch}{1.7}
%\large
%%%%%%
%
%
\newcommand{\sect}[1]{\setcounter{equation}{0}\section{#1}}
\newcommand{\subsect}[1]{\subsection{#1}}
\newcommand{\subsubsect}[1]{\subsubsection{#1}}
\renewcommand{\theequation}{\thesection.\arabic{equation}}
%
\newcommand{\lb}[1]{\label{eq:#1}}
\newcommand{\rf}[1]{(\ref{eq:#1})}
\renewcommand{\thefootnote}{\fnsymbol{footnote}}
%
\newcommand{\ra}{\rightarrow}
\newcommand{\la}{\leftarrow}
\newcommand{\ua}{\uptarrow}
\newcommand{\da}{\downarrow}
\newcommand{\lra}{\longrightarrow}
\newcommand{\lla}{\longleftarrow}
%
\newcommand{\eCD}{\endCD}
\newcommand{\cdr}{\cd\rightarrow}
\newcommand{\cdl}{\cd\leftarrow}
\newcommand{\cdu}{\cd\uptarrow}
\newcommand{\cdd}{\cd\downarrow}
\newcommand{\cdud}{\cd\updowntarrow}
%
%%%%%%%%%%%%%%%%%%%%%% MACROS %%%%%%%%%%%%%%%%%
%
\def\col#1,#2,#3,#4 {\begin{array}{c}#1\\ #2\\ #3\\#4 \end{array}}
\def\prop #1.{\bigskip \noindent {\bf Proposition #1.} \par}
\def\proof{\bigskip \noindent {\it Proof.} \ }
\def\d#1,#2{\frac{d#1}{d#2}}
\def\pd#1,#2{\frac{\partial#1}{\partial#2}}
\def\lpd#1,#2{\frac{\stackrel{\rightarrow}{\partial#1}}{\partial#2}}
\def\rpd#1,#2{\frac{\stackrel{\leftarrow}{\partial#1}}{\partial#2}}
\def\td#1{\dot{#1}}
\def\ttd#1{\ddot{#1}}
\def\tev{\frac{d}{dt}}
\def\fraz#1,#2{\frac{#1}{#2}}
\def\wh{\widehat}
\def\hr{\hat{r}}
\def\vr{\vec{r}}
\def\vq{\vec{q}}
\def\tvr{\td{\vec{r}}}
\def\tvq{\td{\vec{q}}}
\def\bq{{\bf q}}
\def\tbq{\td{\bf q}}
\def\vp{\vec{p}}
\def\wt{\widetilde}
\def\inner{\underline {\ \ }\kern -.1em \raise.3ex\hbox{$\vert $}\,}
\def\ker{\mbox{ker}~}
\def\Hom{\mbox{Hom}~}
\def\im{\mbox{im}~}
\def\Der{\mbox{Der}}
\def\rank{\mbox{rk}}
\def\bsk{\bigskip}
\def\be{\begin{equation}}
\def\ee{\end{equation}}
\def\bea{\begin{eqnarray}}
\def\eea{\end{eqnarray}}
\def\nn{\nonumber}
\def\fl{\forall~}
\def\om{\omega}
\def\de{\delta}
\def\si{\sigma}
\def\th{\theta}
\def\wth{\wt{\th}}
\def\ta{\tau}
\def\De{\Delta}
\def\Ga{\Gamma}
\def\La{\Lambda}
\def\Si{\Sigma}
\def\Om{\Omega}
\def\Laf{\Lambda^{\flat}}
\def\Las{\Lambda^{\sharp}}
\def\thl{\th_{\L}}
\def\oml{\om_{\L}}
%\def\th0{\th_{0}}
%\def\om0{\om_{0}}
%\def\fkg{\frak g}
\def\fkg{\bf g}
\def\mcl{g^{-1}dg}
\def\mcr{(dg)g^{-1}}
\def\mcdl{g^{-1}\td{g}}
\def\mcdr{\td{g} g^{-1}}
\def\me{{\sf m}}
\def\A{{\cal A}}
\def\B{{\cal B}}
\def\C{{\cal C}}
\def\D{{\cal D}}
\def\F{{\cal F}}
\def\G{{\cal G}}
\def\H{{\cal H}}
\def\I{{\cal I}}
\def\L{{\cal L}}
\def\M{{\cal M}}
\def\N{{\cal N}}
\def\O{{\cal O}}
\def\P{{\cal P}}
\def\Q{{\cal Q}}
\def\R{{\cal R}}
\def\S{{\cal S}}
\def\T{{\cal T}}
\def\Z{{\cal Z}}
\def\X{{\cal X}}
\def\FG{\F_\Gamma}
%\renewcommand{\Rb}{\Bbb R}
%\renewcommand{\Cb}{\Bbb C}
%\def\Cb{{\Bbb C}
%\def\Nb{{\Bbb N}}
%\def\Rb{{\Bbb R}}
%\def\Zb{{\Bbb Z}}
\def\Cb{{\bf C}}
\def\Hb{{\bf H}}
\def\Nb{{\bf N}}
\def\Rb{{\bf R}}
\def\Zb{{\bf Z}}
\def\Ab{{\bf A}}
\def\Bb{{\bf B}}
\def\Gb{{\bf G}}
\def\Mb{{\bf M}}
\def\Qb{{\bf Q}}
\def\Tb{{\bf T}}
\def\Xb{{\bf X}}
\def\func{\F(M)}
\def\vect{\X(M)}
\def\form{\X(M)^*}
\def\tens{\T^{r}_{s}(M)}
\def\ham{\X_{\H}(M)}
\def\lham{\X_{\L \H}(M)}
\def\poi{\X_{\P}(M)}
\def\pham{\X_{\P \H}(M)}
\def\plham{\X_{\P \L \H}(M)}
\def\tanb{TQ}
\def\cotb{T^*Q}
\def\funq{\F(Q)}
\def\vecq{\X(Q)}
\def\forq{\X(Q)^*}
\def\tenq{\T^{r}_{s}(Q)}
\def\funtq{\F(\tanb)}
\def\vectq{\X(\tanb)}
\def\fortq{\X(\tanb)^*}
\def\tentq{\T^{r}_{s}(\tanq)}
\def\funcq{\F(\cotb)}
\def\veccq{\X(\cotb)}
\def\forcq{\X(\cotb)^*}
\def\tencq{\T^{r}_{s}(\cotb)}
\def\sfunc{\G(S)}
\def\svect{\X(S)}
\def\sform{\X(S)^*}
\def\stens{\T^{r}_{s}(S)}
\def\hT{\widehat T}
\def\cT{\check T}
\def\NT{{\bf N}_{T}}
\def\HT{{\bf H}_{T}}
\def\GNT{{^{G}\bf N}_{T}}
\def\GHT{{^{G}\bf H}_{T}}
\def\Cinf{C^\infty}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{document}
\thispagestyle{empty}
\hfill July 13 1993
\vspace{.5cm}
\begin{center}
{\Large \bf Recursion Operators: Meaning and Existence
\\~~\\
for Completely Integrable Systems}
\footnote[1]
{Supported in part by the italian Ministero
dell' Universit\`a e della Ricerca Scientifica e Tecnologica.}
\end{center}
\vspace{1.5cm}
\begin{center}
{\Large G. Landi}
\footnote[7]{
Dipartimento di Scienze Matematiche, Universit\`a di Trieste,
P.le Europa, 1, I-34100 Trieste, Italy.~ landi@univ.trieste.it}
\footnotemark[4]
\footnotemark[5]
, {\Large G. Marmo}
\footnote[2]{
Dipartimento di Scienze Fisiche, Universit\`a di Napoli,
Mostra d'Oltremare, Pad.19, I-80125 Napoli, Napoli, Italy.~
gimarmo@na.infn.it}
\footnotemark[4]
, {\Large G. Vilasi}
\footnote[3]{
Dipartimento di Fisica Teorica e SMSA, Universit\`a di Salerno,
Via S. Allende, I-84081 Baronissi (SA), Italy.~vilasi@sa.infn.it}
\footnote[4]{INFN - Sezione di Napoli, Italy.}
\footnotetext[5]
{Fellow of the Italian National Council of
Research (CNR) under Grant 203.01.61.}
\end{center}
\bigskip
\begin{center}
{\it E. Schr\"odinger International Institute for Mathematical Physics, \\
Pasteurgasse 4/7, A-1090 Wien, Austria.}
\end{center}
\bigskip
\vspace{2.5cm}
\begin{abstract}
We show that any non resonant integrable system admits infinitely many
Hamiltonian descriptions and recursion operators.
\vspace{5.0cm}
\end{abstract}
\vspace{8.0cm}
\vfill\eject
\sect{Introduction}
It is by now well known that completely integrable Hamiltonian
dynamical systems
may admit more than one Hamiltonian description
(see for instance [Ma], [GD]).
Usually, with
these alternative descriptions one associates a $(1,1)$ tensor field
which can be used (under suitable conditions) as a recursion operator,
namely as an operator which generate enough constants of the motion in
involution. It seems to be an open question whether it is possible to
find a recursion operator for any completely integrable system.
In the hypotesis of non resonance (see later for the meaning of this
term)
it has been shown that a recursion operator can always be
constructed (even for some infinite dimensional systems) [DMSV].
A recent paper claims however that this is not the case [Br].
It seems to us that it is of some interest to comment on possible
meanings of recursion operators and to show that in condition of non
resonance any integrable system can be reduced to a linear normal
form via a nonlinear non-canonical transformation. For these normal
forms it is straightforward to construct recursion operators. In
particular, we constract one such an operator for the
(counter)example given in [Br].
\bigskip \bigskip
\sect{Integrable Systems (ISs)}\label{se:is}
Let $M$ be a smooth $2n$-dimensional manifold.
Let us suppose we can find $n$ vector fields $X_1, \dots, X_n
\in \vect$ and $n$ functions $F_1, \dots, F_n \in \func$ with the
following properties
\bea
&& [X_i, X_j] = 0~, \lb{cis1} \\
&& L_{X_i} F^j = 0~. ~~~i, j \in \{1, \dots n\}~. \lb{cis2}
\eea
Let us suppose also that on an open dense submanifold of $M$ we have
that
\bea
&& X_1 \wedge \cdots \wedge X_n \not= 0~, \lb{cis4} \\
&& dF^1 \wedge \cdots \wedge dF^n \not= 0~. \lb{cis5}
\eea
We shall show that any dynamical system $\Ga$ on $M$ which is of the
form
\be
\Ga = \sum_{i=1}^{n} \nu^i X_i~,~~~\nu^i = \nu^i(F^1, \dots F^n)~,
\lb{cis3}
\ee
is explicitely integrable on the submanifold on which \rf{cis4} and
\rf{cis5} are satisfied.
We assume finally, that the level sets of the submersion
\be
{\bf F} : M \ra \Rb^n~,~~~ {\bf F} = (F^1, \dots, F^n)~, \lb{cis6}
\ee
are compact. Then the vector fields $X_i$ are complete on each leaf
${\bf F}^{-1}({\bf a}),~ {\bf a} \in \Rb^n$, and they integrate to a
locally free
action of the abelian group $\Rb^n$. Moreover, each leaf is
parallelizable and we can find closed $1$-forms $\alpha^1, \dots,
\alpha^n$, $d\alpha^i = 0$, such that
\be
\alpha^i (X_j) = \delta^i_j~,~~~i, j \in \{1, \dots n\}~.
\lb{cis7}
\ee
With all previous construction, the vector field $\Ga$ in
\rf{cis3} can be explicitely integrated in a neighbourhood of each
leaf ${\bf F}^{-1}({\bf a})$ where we take as coordinates the
functions $\{F^i, \phi^j\}$ with $d\phi^j = \alpha^j$.
The equations of motion of $\Ga$ are given by
\bea
&&\dot \phi^i = \nu^i(F^1, \dots, F^n)~, \nn \\
&&\dot F^i = 0 ~. \lb{cis8}
\eea
Therefore, the corresponding solutions are
\bea
&&\phi_i(t) = t \nu^i({\bf F}(m_0)) + \phi^i(m_0)~, \nn \\
&&F_i(t) = F_i(m_0)~, \lb{cis9}
\eea
with $m_0 \in M$ the initial point. We see that the functions $\nu^i$
play the r\^ole of frequencies.
We stress the fact that up to now we have not used any Hamiltonian
structure.
\bigskip \bigskip
\subsect{Alternative Hamiltonian Descriptions for ISs}\label{se:isah}
We shall now investigate under which conditions a
dynamical system which is
integrable in the sense stated before admits infinitely many
alternative Hamiltonian descriptions.
With $n$-functions $f^1, \dots, f^n$ obeying the
condition $df_i^1 \wedge dF^1 \wedge \cdots \wedge dF^n = 0~,
\forall i \in \{1, \ldots, n \}$,
we can associate a closed $2$-form by setting
\be
\omega_f = \sum_{i} df_i \wedge \alpha^i~, \lb{cis10}
\ee
which is non degenerate as long as
$df_1 \wedge \cdots \wedge df_n \not= 0$.
Any one of these
symplectic $2$-forms makes the action of $\Rb^n$ a Hamiltonian
one. Indeed, by construction of $\om_f$,
\be
i_{X_j}\om_f = - df_j~,~~~j \in \{1,\dots ,n\}~. \lb{cis11}
\ee
As for the vector field $\Ga$ in \rf{cis3} we shall have that
\be
i_{\Ga}\om_f = - \sum_{i} \nu^i d f_i~. \lb{cis12}
\ee
A necessary condition for $i_{\Ga}\omega_F$ to be exact is that
it is closed, namely that
\be
\sum_{i} d\nu^i \wedge d f_i = 0~. \lb{cis13}
\ee
All sets of solutions of this equation for $f^1, \dots, f^n$
satisfying $df_1 \wedge \cdots \wedge df_n \not= 0$ will give
alternative Hamiltonian descriptions for
the dynamical systems $\Ga$ in \rf{cis5}.
Moreover, any such $\Ga$ will be completely integrable in
the Liouville-Arnold
sense, the functions $f_1 ,\dots, f_n$
being constants of the motion (by assumption \rf{cis2}) in involution,
\be
\{f_i , f_j\}_{A} = \omega_{f}(X_i, X_j) = L_{X_i} f_j = 0~.
\lb{cis15}
\ee
\bigskip
There are two limiting case where it is easy to exibit solutions
of \rf{cis13}.
\begin{description}
\item{1.} {\it The constant case.}
All the frequencies $\nu^i$ are constant numbers so that $d\nu^i = 0$ and
\rf{cis13} is automatically satisfied.
\end{description}
Any $2$-form in \rf{cis10} is
an admissible symplectic structure and the corresponding Hamiltonian
function is given by
\be
\om_f = \sum_i \nu^i f_i~. \lb{cis13a}
\ee
An example of system for which
this happens is given by the $n$-dimensional harmonic oscillator
written as
\bea
&& \Ga = \sum_i \nu^i \Ga_i~, \nn \\
&& \Ga_i = \frac{1}{\sqrt{m_ik_i}}p_i \pd{},{q_i} -
\sqrt{m_ik_i} q_i \pd{},{p_i}~,~~~{\rm no~ sum}~, \nn \\
&& \nu^i = \frac{k_i}{m_i}~.\lb{cis15a}
\eea
Here $m_i$ and $k_i$ are the mass and the elastic constant of the
$i$-th oscillator.
Now the functions $F^i$ are just given by the partial Hamiltonians
\be
F^i = \frac{1}{2} ( \frac{p_i^2}{m_i} + k_i q_i^2 )~,
~~~i \in \{1, \dots, n\}~. \lb{cis15b}
\ee
\bigskip
\begin{description}
\item{2.} {\it The non resonant case.}
None of the frequencies $\nu^i$ is constant and we have that
$d\nu^1 \wedge \dots \wedge d\nu^n \not= 0$. In this case we may think of the
$\nu^i$ as `coordinates' and of the $f^j$ as functions of the $\nu^i$.
\end{description}
In this second case, very
simple solutions of \rf{cis13} are given by linear functions
$f_i = \sum_j A_{ij} \nu^j~,~~
i \in \{1, \dots n\}~,~A_{ij} \in \Rb$.
The corresponding Hamiltonian description for $\Ga$ can given with
quadratic Hamiltonian functions by
\bea
&& \om_A = \sum_{ij} A_{ij} d\nu^i \wedge \alpha^j~, \lb{cis16} \\
&& H_A = \frac{1}{2} \sum_{ij} A_{ij}\nu^i \nu^j~. \lb{cis17}
\eea
Moreover, any other symplectic structure of the form
\be
\om_f = \sum_{i} df_i(\nu^i) \wedge \alpha^j~, \lb{cis18}
\ee
in which any $f_i$ depends only on the corresponding frequency
$\nu^i$,
will be admissible as long as $\om_f$ is non degenerate, i.e. as long
as $df_1 \wedge \cdots \wedge df_n \ne 0$.
The associated Hamiltonian functions depend on the explicit form of
the functions $f_i$.
For instance, if $f_i = \pd{G_i},{\nu^i}(\nu^i)$,
the corresponding Hamiltonian can be written as
\be
H_G = \sum_i (G_i - \nu^i \pd{G_i},{\nu^i})~. \lb{cis19}
\ee
\bigskip
A simple example for these case is given again by the $n$-dimensional
harmonic oscillator written as
\be
\Ga = \sum_i F^i \Ga_i~, \lb{cis19a}
\ee
where $F^i$ and $\Ga_i$ are given by \rf{cis15b} and \rf{cis15a}
respectively. Now the partial Hamiltonians $F^i$ play the r\^ole of
frequencies.
\bigskip
\noindent
{\bf Remark.} The intermediate cases are more involved.
For further comments on them we refer to [DMSV].
\bigskip
\noindent
{\bf Remark.} It is worth stressing that there may be admissible
Hamiltonian structures for $\Ga$ which cannot be derived by using
the previous construction.
\bigskip \bigskip
\subsect{Recursion Operators for ISs}\label{se:isro}
We shall now show how to construct recursion operators for the
ISs which we have considered in previous sections. As we have
seen, given the dynamical system \rf{cis3} we can costruct infinitely
many Hamiltonian structures given for instance by \rf{cis10} or
\rf{cis18}.
\begin{description}
\item{1.} {\it The constant case.}
$d\nu^i = 0,~ \forall~ i \in \{1, \dots, n\}$.
\end{description}
Two possible alternative symplectic structures are obtained from
\rf{cis10} as
\bea
&&\om_1 = \sum_{ij} \delta_{ij} dF^i \wedge \alpha^j~
= \sum_k \om_k, \lb{cis20} \\
&&\om_f = \sum_{ij} \delta_{ij} f^i(F^i) dF^i \wedge \alpha^j~
= \sum_k f^k(F^k) \om_k~, \lb{cis21}
\eea
with the condition $df_1 \wedge \cdots \wedge df_n \ne 0$.
Given them, we can construct a $(1,1)$ tensor field $T$ on $M$ by
\be
T = \om_f \circ \om_1^{-1} = \sum_k f^k(F^k)~{\bf I}_k~, \lb{cis21a}
\ee
where ${\bf I}_k$ is the identity operator on the $k$-th two
dimensional `plane' of $T^*M$ with `coordinates' $(dF^k, \alpha^k)$.
\bigskip
\begin{description}
\item{2.} {\it The non resonant case.}
$d\nu^1 \wedge \dots d\nu^n \not= 0$.
\end{description}
In this case two possible alternative symplectic descriptions are
obtained from \rf{cis18} as
\bea
&&\om_1 = \sum_{ij} \delta_{ij} d\nu^i \wedge \alpha^j~
= \sum_k \om_k, \lb{cis22} \\
&&\om_f = \sum_{ij} \delta_{ij} f^i(\nu^i) d\nu^i \wedge \alpha^j~
= \sum_k f^k(\nu^k) \om_k~, \lb{cis23}
\eea
with the condition $df_1 \wedge \cdots \wedge df_n \ne 0$.
Given these structures we can construct a $(1,1)$ tensor field $T$
on $M$ by
\be
T = \om_f \circ \om_1^{-1} = \sum_k f^k(\nu^k)~{\bf I}_k~, \lb{cis24}
\ee
where ${\bf I}_k$ is the identity operator on the $k$-th two
dimensional `plane' of $T^*M$ with `coordinates' $(d\nu^k, \alpha^k)$.
>From the way they are constructed, one sees that $T$ in \rf{cis21} and
\rf{cis24} are invariant under
$\Ga$, have double degenerate spectrum with eigenfunctions without
critical points, and vanishing Nijenhuis torsion $\N_T$
\footnote{We remind that the tensor $\N_T$ is defined by
$\N_T(X, Y) = [TX, TY] - T[TX, Y] - T[X, TY] + T^2[X, Y]~,~
\forall~ X, Y \in \vect$.}.
Therefore they are
recursion operators for the dynamical system $\Ga$.
\bigskip \bigskip
\subsect{Liouville-Arnold ISs}\label{se:alis}
Assume the dynamical vector field $\Ga$ on the symplectic manifold
$(M, \om_0)$ has $n$ constants of the
motion $H^1, \dots, H^n$ which are in involution (with respect to the
Poisson structure associated with $\om_0$), functionally
independent, $dH^1\wedge \cdots \wedge dH^n \ne 0$, and generate
complete vector fields $X_1,\dots,X_n$.
We have then an action of $\Rb^n$ on $M$ which is locally free and
fibrating
In this situation one finds `{\it angle}' 1-forms
$\alpha^1, \dots, \alpha^n$
such that $\alpha^i(X_j) = \delta^i_j$ and $d\alpha^i = 0$.
Given any function $F$ of the $H^j$,
(or $dF \wedge dH^1 \wedge \cdots \wedge dH^n = 0$)
such that $det||\pd {^2 F}, {{H^i}{\partial {H^j}}}||\ne 0$,
the 2-form
\be
\omega_F = d( {\pd F,{H^i}} \alpha^i)
\ee
is an admissible symplectic structure for the $\Rb^n$ action.
In particular, if
\be
F = {1\over 2}\sum_i H^2_i
\ee
we just get back the $\{ H^i \}$ as Hamiltonian functions.
With a set of {\it action-angles} variables $(J_k, \phi^k)$ we have
that
\bea
&&\Gamma = \nu^k \pd{},{\phi^k}~,\\
&&\omega_0 = dJ_k \wedge d\phi^k~,\\
&&i_{\Gamma}\omega = \nu^k dJ_k = - \pd{H},{J_k} dJ_k = -dH~,
\eea
where $\nu^k = \pd{H},{J_k}~, ~k \in \{1, \dots, n\}$ are the frequencies.
In the {\it non resonant case} when
$d \nu^1 \wedge \cdots\wedge d\nu^n \not= 0$ or equivalently,
$det|| \pd{\nu^k},{J_l}|| \ne 0$,
\footnote{This is also equivalent to the notion of
nondegeneracy of the Hamiltonian function used in [Br].}
we can use the
$\nu^k$ as variables and write the admissible symplectic structure
\be
\omega_{\nu} = \sum_k d\nu^k \wedge d \phi^k~,
\ee
with Hamiltonian a quadratic function
\be
H_{\nu} = \frac{1}{2} \sum_k (\nu^k)^2.
\ee
By using the analysis of section \ref{se:isah}
we obtain that a not resonant complete integrable system has
infinitely many admissible symplectic structures, some of them having
the form
\be
\omega_f = \sum_i df_i(\nu^i) \wedge d\phi^i~,
\ee
with the condition $df^1 \wedge \cdots df^n \not= 0$.
However, in general, we may not obtain $\om_0$ in this way.
Moreover, such systems do admit recursion operators given by
expression \rf{cis24}.
\bigskip \bigskip
\subsect{The example of Brouzet}
In [Br] the following $2$-degrees of freedom, completely integrable
system is considered. Take $M = \Rb^2 \times \Tb^2 = \{(x, y, \theta,
\eta)\}$ with symplectic structure $\om_0 = dx \wedge d\theta + dy
\wedge d \eta $. The dynamical system is described by the Hamiltonian
$H = x^3 + y^3 + xy$. The corresponding dynamical vector field is
given by
\bea
&& \Ga = \nu_{\theta} \pd{},{\theta} + \nu_{\eta} \pd{},{\eta}~,
\nn \\
&& \nu_{\theta} = 3 x^2 + y~, \nn \\
&& \nu_{\eta} = 3 y^2 + x~. \lb{br1}
\eea
>From what we have said before this system admits infinitely may
alternative Hamiltonian descriptions in the dense open submanifold
characterized by $ d\nu_{\theta} \wedge d\nu_{\eta} \not= 0$,
namely by $36 x y - 1 \not= 0$, which coincides with the submanifold
on which $H$ is nondegenerate. Two such structures are given by
\bea
&&\om_1 = d\nu_{\theta} \wedge d\theta + d\nu_{\eta} \wedge d\eta~, \lb{br2}
\\
&&\om_2 = f(\nu_{\th}) d\nu_{\theta} \wedge d\theta
+ g(\nu_{\eta}) d\nu_{\eta} \wedge d\eta~, \lb{br3}
\eea
where $f$ and $g$ are any two functions such that
$df \wedge dg \not= 0$. The corresponding recursion operators are given
by
\be
T = \om_2 \circ \om_1^{-1} =
f(\nu_{\th}) \left( d\nu_\th \otimes \pd{},{\nu_\th} +
d\th \otimes \pd{},{\th} \right)
+ g(\nu_{\eta}) \left(d\nu_\eta \otimes \pd{},{\nu_\eta}
+ d\eta \otimes \pd{},{\eta} \right)~. \lb{br4}
\ee
We stress the fact that $\om_0$ is not among the symplectic structures
constructed in \rf{br3} and that our recursion operators \rf{br4}
cannot be `factorized' through $\om_0$.
\bigskip
>From this exemple it is clear that there is some ambiguity on what is
a recursion operator for a given dynamical system. In the coming section
we would like to make more clear this point.
\bigskip \bigskip
\sect{Recursion operators}
We notice that in studying the integrability of a given system $\Ga$
we could start directly with a $(1, 1)$ tensor field $T$ satisfying
$L_{\Ga}T = 0$, with double degenerate spectrum, with eigenfunctions
without critical points, and vanishing Nijenhuis torsion $\N_T$.
It is not necessary to
require that $T$ be factorizable via symplectic structures. Indeed
this non unique decomposition can be constructed afterwards [DMSV].
In this section we shall comment some more on the meaning of recursion
operators and on their use in the analysis of complete integrability
[ZC], [Mar].
Therefore let us suppose we have a dynamical vector field $\Ga \in \vect$ and a
compatible $(1, 1)$ tensor $T$~, namely $L_\Ga T = 0$~, so that the
functions $trT^k~, ~k \ge 1$ are constants of the motion. By applying
powers of $T$ we get vector fields $\Ga_k = T^k(\Ga)$~ which are
symmetries for $\Ga$. The Lie algebra $\{ \Ga_k~,~ k \ge 0 \}$
is abelian if $\N_T = 0$.
\bigskip
If $F \in \func$ is a constant of the motion for $\Ga$, we say that
$T$ is an {\it $F$-weak recursion operator} if
$\N_T = 0$ and $d(T(dF)) = 0$~
(we use the same symbol for $T$ and for its dual).
If $T$ is an $F$-weak recursion operator, one can prove that
$d(T^k(dF)) = 0~,~ \forall ~k > 1$~. Locally, one finds functions
$F_k \in \func$ by $dF_k = T^k (dF)$ which are constants of the motion
for $\Ga$.
It is worth stressing that a given operator $T$ may be a recursion
operator for the constant of the motion $F$ and not a recursion for
another constant of the motion $G$. Moreover, it may also happen that
the tensor $T$ is an $F$-recursion operator but ~$T^k(dF) \wedge dF = 0~,~
\forall~ k \ge 1$, so that one cannot use $T$ and $F$
to generate new constants of the
motion.
This is what happens for instance with the Kepler problem if one
starts with the standard Hamiltonian function [MV].
However, it is always true that
$T(d( \frac{1}{k} tr T^k)) = d ( \frac{1}{k+1} tr T^{k+1}).$
\bigskip
If $\om$ is an admissible symplectic structure for $\Ga$, namely
$L_\Ga \om = 0$, we say that
$T$ is an {\it $\om$-weak recursion operator} if $\N_T = 0$ and
$d(T(\om)) = 0$~
(again, we use the same symbol for the extension of
$T$ to forms).
If $T$ is an $\om$-weak recursion operator one proves that
$d(T^k(\om)) = 0~,~ \forall~ k > 1~$. All $2$-forms $\om_k =
T^k(\om)$ are then admissible symplectic structures for $\Ga$.
It is worth stressing that given any two admissible symplectic
structures $\om_1$ and $\om_2$ for $\Ga$, it need not be true that
they are connected by a recursion operator. Moreover, it may happen
that $T^k(\om) \wedge \om = 0~,~ \forall~ k \ge 1$~ so that one does
not generate new symplectic structures.
\bigskip
If $\Ga$ is Hamiltonian with respect to the couple $(\om, H)$, namely
$i_\Ga \om = -dH$, we say that $T$ is a {\it strong recursion
operator} if it is both an $H$-recursion operator and an
$\om$-recursion operator. If this is the case, any vector field
$\Ga_k$ is an Hamiltonian one with respect to $\om$ with Hamiltonian
function $H_k$ as well as with respect to $\om_k$ with Hamiltonian function
$H$. Moreover, the constants of the motion $H_k$ are pairwise in
involution with respect to the Poisson structure constructed by
inverting anyone of the symplectic structures $\om_k~, ~k \ge 0$.
\bigskip \bigskip
\sect{Conclusions}
We have shown that any non resonant integrable system admits
infinitely many alternative symplectic structures and strong
recursion operators.
This class of systems include Hamiltonian ones with non degenerate
Hamiltonian functions.
The result proven in [Br] seems rather to exlude the possibility of
constructing recursion operators while {\it keeping fixed} one
symplectic structure $\om_0$, namely to construct
$\om_0$-recursion operator.
But this not surprising result seems less
relevant for the analysis of complete integrable systems.
\bigskip \bigskip \bigskip
\vfill\eject
\noindent
{\bf REFERENCES.}
\begin{description}
\item[Br] R. Brouzet, {\it About the existence of recursion operators
for completely integrable Hamiltonian systems near a Liouville
torus}, J. Math. Phys. {\bf 34} (1993) 1309-1313.
\item[DMSV] S. De Filippo, G. Marmo, M. Salerno, G. Vilasi,
{\it On the Phase Manifold Geometry of Integrable Nonlinear Field
Theory}, Preprint IFUSA, Salerno (1982), unpublished.
{\it A new Characterization of Complete Integrable Systems}
Il N. Cimento {\bf 83B} (1984) 97-112.
\item[GD] I. M. Gelfand, I. Ya. Dorfman,
{\it The Schouten Bracket and Hamiltonian Operators}, Funct. Anal.
Appl. {\bf 14} (1980) 71-74.
\item[Ma] F. Magri, {\it A simple model of the integrable Hamiltonian
equation}, J. Math. Phys. {\bf 19} (1978) 1156-62.
\item[Mar] G. Marmo, {\it Nijenhuis Operators in Classical Dynamics},
in `Proceedings of Seminar on Group Theoretical Methods in Physics',
USSR Academy of Sciences, Yurmala, Latvian SSR, May 1985.
\item[MV] G. Marmo, G. Vilasi, {\it When do recursion operators
generate new conservation laws?}, Phys. Lett. {\bf B 277} (1992)
137-140.
\item[ZK] V. E. Zakharov, B. G. Konopelchenko, {\it On the Theory of
Recursion Operator}, Commun. Math. Phys. {\bf 94} (1984) 483-509.
\end{description}
\end{document}